ANSYS FLUENT Theory Guide

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Table of ContentsUsing This Manual ..................................................................................................................................... xxv 1. The Contents of This Manual ............................................................................................................. xxv 2. The Contents of the FLUENT Manuals ............................................................................................... xxvi 3. Typographical Conventions ............................................................................................................ xxvii 4. Mathematical Conventions ............................................................................................................. xxvii 5. Technical Support ........................................................................................................................... xxix 1. Basic Fluid Flow ....................................................................................................................................... 1 1.1. Overview of Physical Models in ANSYS FLUENT .................................................................................. 1 1.2. Continuity and Momentum Equations ............................................................................................... 2 1.2.1. The Mass Conservation Equation .............................................................................................. 2 1.2.2. Momentum Conservation Equations ........................................................................................ 3 1.3. User-Defined Scalar (UDS) Transport Equations .................................................................................. 4 1.3.1. Single Phase Flow .................................................................................................................... 4 1.3.2. Multiphase Flow ....................................................................................................................... 5 1.4. Periodic Flows .................................................................................................................................. 6 1.4.1. Overview ................................................................................................................................. 7 1.4.2. Limitations ............................................................................................................................... 7 1.4.3. Physics of Periodic Flows .......................................................................................................... 8 1.4.3.1. Definition of the Periodic Velocity .................................................................................... 8 1.4.3.2. Definition of the Streamwise-Periodic Pressure ................................................................ 8 1.5. Swirling and Rotating Flows .............................................................................................................. 9 1.5.1. Overview of Swirling and Rotating Flows ................................................................................ 10 1.5.1.1. Axisymmetric Flows with Swirl or Rotation ..................................................................... 10 1.5.1.1.1. Momentum Conservation Equation for Swirl Velocity ............................................. 11 1.5.1.2. Three-Dimensional Swirling Flows .................................................................................. 11 1.5.1.3. Flows Requiring a Moving Reference Frame ................................................................... 11 1.5.2. Physics of Swirling and Rotating Flows .................................................................................... 11 1.6. Compressible Flows ........................................................................................................................ 12 1.6.1. When to Use the Compressible Flow Model ............................................................................ 14 1.6.2. Physics of Compressible Flows ................................................................................................ 14 1.6.2.1. Basic Equations for Compressible Flows ......................................................................... 15 1.6.2.2. The Compressible Form of the Gas Law .......................................................................... 15 1.7. Inviscid Flows ................................................................................................................................. 16 1.7.1. Euler Equations ...................................................................................................................... 16 1.7.1.1. The Mass Conservation Equation .................................................................................... 16 1.7.1.2. Momentum Conservation Equations .............................................................................. 17 1.7.1.3. Energy Conservation Equation ....................................................................................... 17 2. Flows with Moving Reference Frames ................................................................................................... 19 2.1. Introduction ................................................................................................................................... 19 2.2. Flow in a Moving Reference Frame .................................................................................................. 21 2.2.1. Equations for a Moving Reference Frame ................................................................................ 21 2.2.1.1. Relative Velocity Formulation ......................................................................................... 22 2.2.1.2. Absolute Velocity Formulation ....................................................................................... 23 2.2.1.3. Relative Specification of the Reference Frame Motion ..................................................... 24 2.3. Flow in Multiple Reference Frames .................................................................................................. 24 2.3.1. The Multiple Reference Frame Model ...................................................................................... 25 2.3.1.1. Overview ....................................................................................................................... 25 2.3.1.2. Examples ....................................................................................................................... 25 2.3.1.3. The MRF Interface Formulation ...................................................................................... 27 2.3.1.3.1. Interface Treatment: Relative Velocity Formulation ................................................. 27Release 14.0 - SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.

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ANSYS FLUENT Theory Guide 2.3.1.3.2. Interface Treatment: Absolute Velocity Formulation ............................................... 28 2.3.2. The Mixing Plane Model ......................................................................................................... 28 2.3.2.1. Overview ....................................................................................................................... 29 2.3.2.2. Rotor and Stator Domains .............................................................................................. 29 2.3.2.3. The Mixing Plane Concept ............................................................................................. 30 2.3.2.4. Choosing an Averaging Method ..................................................................................... 31 2.3.2.4.1. Area Averaging ..................................................................................................... 31 2.3.2.4.2. Mass Averaging .................................................................................................... 31 2.3.2.4.3. Mixed-Out Averaging ............................................................................................ 32 2.3.2.5. Mixing Plane Algorithm of ANSYS FLUENT ..................................................................... 33 2.3.2.6. Mass Conservation ........................................................................................................ 33 2.3.2.7. Swirl Conservation ......................................................................................................... 33 2.3.2.8. Total Enthalpy Conservation .......................................................................................... 34 3. Flows Using Sliding and Dynamic Meshes ............................................................................................ 35 3.1. Introduction ................................................................................................................................... 35 3.2. Dynamic Mesh Theory .................................................................................................................... 36 3.2.1. Conservation Equations ......................................................................................................... 37 3.2.2. Six DOF (6DOF) Solver Theory ................................................................................................. 38 3.3. Sliding Mesh Theory ....................................................................................................................... 39 4. Turbulence ............................................................................................................................................. 41 4.1. Underlying Principles of Turbulence Modeling ................................................................................. 41 4.1.1. Reynolds (Ensemble) Averaging .............................................................................................. 41 4.1.2. Filtered Navier-Stokes Equations ............................................................................................. 42 4.1.3. Hybrid RANS-LES Formulations ............................................................................................... 44 4.1.4. Boussinesq Approach vs. Reynolds Stress Transport Models ..................................................... 44 4.2. Spalart-Allmaras Model ................................................................................................................... 44 4.2.1. Overview ............................................................................................................................... 45 4.2.2. Transport Equation for the Spalart-Allmaras Model ................................................................. 45 4.2.3. Modeling the Turbulent Viscosity ............................................................................................ 46 4.2.4. Modeling the Turbulent Production ........................................................................................ 46 4.2.5. Modeling the Turbulent Destruction ....................................................................................... 47 4.2.6. Model Constants .................................................................................................................... 48 4.2.7. Wall Boundary Conditions ...................................................................................................... 48 4.2.8. Convective Heat and Mass Transfer Modeling .......................................................................... 49 4.3. Standard, RNG, and Realizable k- Models ....................................................................................... 49 4.3.1. Standard k- Model ............................................................................................................... 49 4.3.1.1. Overview ....................................................................................................................... 49 4.3.1.2. Transport Equations for the Standard k- Model ............................................................. 50 4.3.1.3. Modeling the Turbulent Viscosity ................................................................................... 50 4.3.1.4. Model Constants ........................................................................................................... 51 4.3.2. RNG k- Model ....................................................................................................................... 51 4.3.2.1. Overview ....................................................................................................................... 51 4.3.2.2. Transport Equations for the RNG k- Model .................................................................... 51 4.3.2.3. Modeling the Effective Viscosity ..................................................................................... 52 4.3.2.4. RNG Swirl Modification .................................................................................................. 53 4.3.2.5. Calculating the Inverse Effective Prandtl Numbers .......................................................... 53 4.3.2.6. The R- Term in the Equation ........................................................................................ 53 4.3.2.7. Model Constants ........................................................................................................... 54 4.3.3. Realizable k- Model .............................................................................................................. 54 4.3.3.1. Overview ....................................................................................................................... 54 4.3.3.2. Transport Equations for the Realizable k- Model ........................................................... 56 4.3.3.3. Modeling the Turbulent Viscosity ................................................................................... 57Release 14.0 - SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.

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ANSYS FLUENT Theory Guide 4.3.3.4. Model Constants ........................................................................................................... 58 4.3.4. Modeling Turbulent Production in the k- Models .................................................................. 58 4.3.5. Effects of Buoyancy on Turbulence in the k- Models .............................................................. 59 4.3.6. Effects of Compressibility on Turbulence in the k- Models ..................................................... 60 4.3.7. Convective Heat and Mass Transfer Modeling in the k- Models .............................................. 60 4.4. Standard and SST k- Models ......................................................................................................... 62 4.4.1. Standard k- Model ............................................................................................................... 62 4.4.1.1. Overview ....................................................................................................................... 62 4.4.1.2. Transport Equations for the Standard k- Model ............................................................ 63 4.4.1.3. Modeling the Effective Diffusivity ................................................................................... 63 4.4.1.3.1. Low-Reynolds-Number Correction ........................................................................ 63 4.4.1.4. Modeling the Turbulence Production ............................................................................. 64 4.4.1.4.1. Production of k ..................................................................................................... 64 4.4.1.4.2. Production of ..................................................................................................... 64 4.4.1.5. Modeling the Turbulence Dissipation ............................................................................. 65 4.4.1.5.1. Dissipation of k ..................................................................................................... 65 4.4.1.5.2. Dissipation of ..................................................................................................... 66 4.4.1.5.3. Compressibility Correction .................................................................................... 67 4.4.1.6. Model Constants ........................................................................................................... 67 4.4.2. Shear-Stress Transport (SST) k- Model .................................................................................. 68 4.4.2.1. Overview ....................................................................................................................... 68 4.4.2.2. Transport Equations for the SST k- Model .................................................................... 68 4.4.2.3. Modeling the Effective Diffusivity ................................................................................... 69 4.4.2.4. Modeling the Turbulence Production ............................................................................. 70 4.4.2.4.1. Production of k ..................................................................................................... 70 4.4.2.4.2. Production of ..................................................................................................... 70 4.4.2.5. Modeling the Turbulence Dissipation ............................................................................. 71 4.4.2.5.1. Dissipation of k ..................................................................................................... 71 4.4.2.5.2. Dissipation of ..................................................................................................... 71 4.4.2.6. Cross-Diffusion Modification .......................................................................................... 72 4.4.2.7. Model Constants ........................................................................................................... 72 4.4.3. Turbulence Damping .............................................................................................................. 72 4.4.4. Wall Boundary Conditions ...................................................................................................... 73 4.5. k- kl- Transition Model .................................................................................................................. 74 4.5.1. Overview ............................................................................................................................... 74 4.5.2. Transport Equations for the k- kl- Model ............................................................................... 74 4.5.2.1. Model Constants ........................................................................................................... 79 4.6. Transition SST Model ....................................................................................................................... 79 4.6.1. Overview ............................................................................................................................... 80 4.6.2. Transport Equations for the Transition SST Model .................................................................... 80 4.6.2.1. Separation Induced Transition Correction ....................................................................... 81 4.6.2.2. Coupling the Transition Model and SST Transport Equations ........................................... 83 4.6.2.3. Transition SST and Rough Walls ...................................................................................... 84 4.6.3. Specifying Inlet Turbulence Levels .......................................................................................... 84 4.7. The V2F Model ................................................................................................................................ 86 4.8. Reynolds Stress Model (RSM) ........................................................................................................... 87 4.8.1. Overview ............................................................................................................................... 87 4.8.2. Reynolds Stress Transport Equations ....................................................................................... 87 4.8.3. Modeling Turbulent Diffusive Transport .................................................................................. 88 4.8.4. Modeling the Pressure-Strain Term ......................................................................................... 89 4.8.4.1. Linear Pressure-Strain Model .......................................................................................... 89 4.8.4.2. Low-Re Modifications to the Linear Pressure-Strain Model .............................................. 90Release 14.0 - SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.

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ANSYS FLUENT Theory Guide 4.8.4.3. Quadratic Pressure-Strain Model .................................................................................... 91 4.8.4.4. Low-Re Stress-Omega Model ......................................................................................... 92 4.8.5. Effects of Buoyancy on Turbulence ......................................................................................... 93 4.8.6. Modeling the Turbulence Kinetic Energy ................................................................................. 94 4.8.7. Modeling the Dissipation Rate ................................................................................................ 95 4.8.8. Modeling the Turbulent Viscosity ............................................................................................ 95 4.8.9. Wall Boundary Conditions ...................................................................................................... 96 4.8.10. Convective Heat and Mass Transfer Modeling ........................................................................ 96 4.9. Scale-Adaptive Simulation (SAS) Model ........................................................................................... 97 4.9.1. Overview ............................................................................................................................... 97 4.9.2. Transport Equations for the SAS Model ................................................................................... 98 4.10. Detached Eddy Simulation (DES) ................................................................................................. 100 4.10.1. Overview ........................................................................................................................... 101 4.10.2. Spalart-Allmaras Based DES Model ...................................................................................... 101 4.10.3. Realizable k- Based DES Model ......................................................................................... 102 4.10.4. SST k- Based DES Model ................................................................................................... 103 4.10.5. Improved Delayed Detached Eddy Simulation (IDDES) ........................................................ 104 4.10.5.1. Overview of IDDES ..................................................................................................... 104 4.10.5.2. IDDES Model Formulation .......................................................................................... 104 4.11. Large Eddy Simulation (LES) Model .............................................................................................. 105 4.11.1. Overview ........................................................................................................................... 105 4.11.2. Subgrid-Scale Models ......................................................................................................... 106 4.11.2.1. Smagorinsky-Lilly Model ............................................................................................ 107 4.11.2.2. Dynamic Smagorinsky-Lilly Model .............................................................................. 108 4.11.2.3. Wall-Adapting Local Eddy-Viscosity (WALE) Model ...................................................... 109 4.11.2.4. Algebraic Wall-Modeled LES Model (WMLES) .............................................................. 110 4.11.2.4.1. Algebraic WMLES Model Formulation ................................................................ 111 4.11.2.4.1.1. Reynolds Number Scaling ......................................................................... 111 4.11.2.5. Dynamic Kinetic Energy Subgrid-Scale Model ............................................................. 113 4.11.3. Inlet Boundary Conditions for the LES Model ....................................................................... 113 4.11.3.1. Vortex Method ........................................................................................................... 114 4.11.3.2. Spectral Synthesizer ................................................................................................... 115 4.12. Embedded Large Eddy Simulation (ELES) ..................................................................................... 116 4.12.1. Overview ........................................................................................................................... 116 4.12.2. Selecting a Model ............................................................................................................... 116 4.12.3. Interfaces Treatment ........................................................................................................... 117 4.12.3.1. RANS-LES Interface .................................................................................................... 117 4.12.3.2. LES-RANS Interface .................................................................................................... 118 4.12.3.3. Internal Interface Without LES Zone ........................................................................... 118 4.12.3.4. Grid Generation Guidelines ....................................................................................... 119 4.13. Near-Wall Treatments for Wall-Bounded Turbulent Flows .............................................................. 119 4.13.1. Overview ........................................................................................................................... 119 4.13.1.1. Wall Functions vs. Near-Wall Model ............................................................................. 120 4.13.1.2. Wall Functions ........................................................................................................... 122 4.13.2. Standard Wall Functions ..................................................................................................... 122 4.13.2.1. Momentum ............................................................................................................... 122 4.13.2.2. Energy ....................................................................................................................... 123 4.13.2.3. Species ...................................................................................................................... 125 4.13.2.4. Turbulence ................................................................................................................ 126 4.13.3. Scalable Wall Functions ...................................................................................................... 127 4.13.4. Non-Equilibrium Wall Functions .......................................................................................... 127 4.13.4.1. Standard Wall Functions vs. Non-Equilibrium Wall Functions ....................................... 129Release 14.0 - SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.

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ANSYS FLUENT Theory Guide 4.13.4.2. Limitations of the Wall Function Approach ................................................................. 129 4.13.5. Enhanced Wall Treatment -Equation (EWT-) ...................................................................... 129 4.13.5.1. Two-Layer Model for Enhanced Wall Treatment ........................................................... 130 4.13.5.2. Enhanced Wall Treatment for Momentum and Energy Equations ................................ 132 4.13.6. Enhanced Wall Treatment -Equation (EWT-) .................................................................... 135 4.13.7. User-Defined Wall Functions ............................................................................................... 135 4.13.8. LES Near-Wall Treatment ..................................................................................................... 135 4.14. Curvature Correction for the Spalart-Allmaras and Two-Equation Models ..................................... 136 5. Heat Transfer ....................................................................................................................................... 139 5.1. Introduction ................................................................................................................................. 139 5.2. Modeling Conductive and Convective Heat Transfer ...................................................................... 139 5.2.1. Heat Transfer Theory ............................................................................................................. 139 5.2.1.1. The Energy Equation .................................................................................................... 139 5.2.1.2. The Energy Equation in Moving Reference Frames ........................................................ 140 5.2.1.3. The Energy Equation for the Non-Premixed Combustion Model .................................... 141 5.2.1.4. Inclusion of Pressure Work and Kinetic Energy Terms .................................................... 141 5.2.1.5. Inclusion of the Viscous Dissipation Terms .................................................................... 141 5.2.1.6. Inclusion of the Species Diffusion Term ........................................................................ 142 5.2.1.7. Energy Sources Due to Reaction ................................................................................... 142 5.2.1.8. Energy Sources Due To Radiation ................................................................................. 143 5.2.1.9. Interphase Energy Sources ........................................................................................... 143 5.2.1.10. Energy Equation in Solid Regions ............................................................................... 143 5.2.1.11. Anisotropic Conductivity in Solids .............................................................................. 143 5.2.1.12. Diffusion at Inlets ....................................................................................................... 143 5.2.2. Natural Convection and Buoyancy-Driven Flows Theory ........................................................ 144 5.3. Modeling Radiation ...................................................................................................................... 144 5.3.1. Overview and Limitations ..................................................................................................... 145 5.3.1.1. Advantages and Limitations of the DTRM ..................................................................... 145 5.3.1.2. Advantages and Limitations of the P-1 Model ............................................................... 146 5.3.1.3. Advantages and Limitations of the Rosseland Model .................................................... 146 5.3.1.4. Advantages and Limitations of the DO Model ............................................................... 146 5.3.1.5. Advantages and Limitations of the S2S Model .............................................................. 147 5.3.2. Radiative Transfer Equation .................................................................................................. 147 5.3.3. P-1 Radiation Model Theory .................................................................................................. 149 5.3.3.1. The P-1 Model Equations ............................................................................................. 149 5.3.3.2. Anisotropic Scattering ................................................................................................. 150 5.3.3.3. Particulate Effects in the P-1 Model .............................................................................. 151 5.3.3.4. Boundary Condition Treatment for the P-1 Model at Walls ............................................. 152 5.3.3.5. Boundary Condition Treatment for the P-1 Model at Flow Inlets and Exits ...................... 153 5.3.4. Rosseland Radiation Model Theory ....................................................................................... 154 5.3.4.1. The Rosseland Model Equations ................................................................................... 154 5.3.4.2. Anisotropic Scattering ................................................................................................. 154 5.3.4.3. Boundary Condition Treatment at Walls ........................................................................ 155 5.3.4.4. Boundary Condition Treatment at Flow Inlets and Exits ................................................. 155 5.3.5. Discrete Transfer Radiation Model (DTRM) Theory ................................................................. 155 5.3.5.1. The DTRM Equations .................................................................................................... 155 5.3.5.2. Ray Tracing .................................................................................................................. 156 5.3.5.3. Clustering .................................................................................................................... 157 5.3.5.4. Boundary Condition Treatment for the DTRM at Walls ................................................... 158 5.3.5.5. Boundary Condition Treatment for the DTRM at Flow Inlets and Exits ............................ 158 5.3.6. Discrete Ordinates (DO) Radiation Model Theory ................................................................... 158 5.3.6.1. The DO Model Equations ............................................................................................. 159Release 14.0 - SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.

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ANSYS FLUENT Theory Guide 5.3.6.2. Energy Coupling and the DO Model ............................................................................. 160 5.3.6.2.1. Limitations of DO/Energy Coupling ..................................................................... 161 5.3.6.3. Angular Discretization and Pixelation ........................................................................... 161 5.3.6.4. Anisotropic Scattering ................................................................................................. 164 5.3.6.5. Particulate Effects in the DO Model .............................................................................. 165 5.3.6.6. Boundary and Cell Zone Condition Treatment at Opaque Walls ..................................... 165 5.3.6.6.1. Gray Diffuse Walls ............................................................................................... 167 5.3.6.6.2. Non-Gray Diffuse Walls ........................................................................................ 167 5.3.6.7. Cell Zone and Boundary Condition Treatment at Semi-Transparent Walls ...................... 168 5.3.6.7.1. Semi-Transparent Interior Walls ........................................................................... 168 5.3.6.7.2. Specular Semi-Transparent Walls ......................................................................... 169 5.3.6.7.3. Diffuse Semi-Transparent Walls ............................................................................ 172 5.3.6.7.4. Partially Diffuse Semi-Transparent Walls ............................................................... 173 5.3.6.7.5. Semi-Transparent Exterior Walls ........................................................................... 173 5.3.6.7.6. Limitations .......................................................................................................... 175 5.3.6.7.7. Solid Semi-Transparent Media ............................................................................. 176 5.3.6.8. Boundary Condition Treatment at Specular Walls and Symmetry Boundaries ................. 176 5.3.6.9. Boundary Condition Treatment at Periodic Boundaries ................................................. 176 5.3.6.10. Boundary Condition Treatment at Flow Inlets and Exits ............................................... 176 5.3.7. Surface-to-Surface (S2S) Radiation Model Theory .................................................................. 176 5.3.7.1. Gray-Diffuse Radiation ................................................................................................. 176 5.3.7.2. The S2S Model Equations ............................................................................................. 177 5.3.7.3. Clustering .................................................................................................................... 178 5.3.7.3.1. Clustering and View Factors ................................................................................ 178 5.3.7.3.2. Clustering and Radiosity ...................................................................................... 179 5.3.8. Radiation in Combusting Flows ............................................................................................ 179 5.3.8.1. The Weighted-Sum-of-Gray-Gases Model ..................................................................... 179 5.3.8.1.1. When the Total (Static) Gas Pressure is Not Equal to 1 atm .................................... 181 5.3.8.2. The Effect of Soot on the Absorption Coefficient ........................................................... 181 5.3.8.3. The Effect of Particles on the Absorption Coefficient ..................................................... 182 5.3.9. Choosing a Radiation Model ................................................................................................. 182 5.3.9.1. External Radiation ....................................................................................................... 183 6. Heat Exchangers .................................................................................................................................. 185 6.1. The Macro Heat Exchanger Models ................................................................................................ 185 6.1.1. Overview of the Macro Heat Exchanger Models .................................................................... 185 6.1.2. Restrictions of the Macro Heat Exchanger Models ................................................................. 186 6.1.3. Macro Heat Exchanger Model Theory .................................................................................... 187 6.1.3.1. Streamwise Pressure Drop ........................................................................................... 188 6.1.3.2. Heat Transfer Effectiveness ........................................................................................... 190 6.1.3.3. Heat Rejection ............................................................................................................. 190 6.1.3.4. Macro Heat Exchanger Group Connectivity .................................................................. 192 6.2. The Dual Cell Model ...................................................................................................................... 193 6.2.1. Overview of the Dual Cell Model ........................................................................................... 193 6.2.2. Restrictions of the Dual Cell Model ........................................................................................ 194 6.2.3. Dual Cell Model Theory ......................................................................................................... 194 6.2.3.1. NTU Relations .............................................................................................................. 195 6.2.3.2. Heat Rejection ............................................................................................................. 195 7. Species Transport and Finite-Rate Chemistry ..................................................................................... 197 7.1. Volumetric Reactions .................................................................................................................... 197 7.1.1. Species Transport Equations ................................................................................................. 197 7.1.1.1. Mass Diffusion in Laminar Flows ................................................................................... 198 7.1.1.2. Mass Diffusion in Turbulent Flows ................................................................................ 198Release 14.0 - SAS IP, Inc. 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ANSYS FLUENT Theory Guide 7.1.1.3. Treatment of Species Transport in the Energy Equation ................................................. 198 7.1.1.4. Diffusion at Inlets ......................................................................................................... 199 7.1.2. The Generalized Finite-Rate Formulation for Reaction Modeling ............................................ 199 7.1.2.1. The Laminar Finite-Rate Model ..................................................................................... 199 7.1.2.2. Pressure-Dependent Reactions .................................................................................... 202 7.1.2.3. The Eddy-Dissipation Model ......................................................................................... 204 7.1.2.4. The Eddy-Dissipation Model for LES ............................................................................. 205 7.1.2.5. The Eddy-Dissipation-Concept (EDC) Model ................................................................. 206 7.1.2.6. The Thickened Flame Model ......................................................................................... 207 7.1.2.7. The Relaxation to Chemical Equilibrium Model ............................................................. 209 7.2. Wall Surface Reactions and Chemical Vapor Deposition .................................................................. 210 7.2.1. Surface Coverage Reaction Rate Modification ....................................................................... 212 7.2.2. Reaction-Diffusion Balance for Surface Chemistry ................................................................. 213 7.2.3. Slip Boundary Formulation for Low-Pressure Gas Systems ..................................................... 214 7.3. Particle Surface Reactions ............................................................................................................. 216 7.3.1. General Description .............................................................................................................. 216 7.3.2. ANSYS FLUENT Model Formulation ....................................................................................... 217 7.3.3. Extension for Stoichiometries with Multiple Gas Phase Reactants .......................................... 219 7.3.4. Solid-Solid Reactions ............................................................................................................ 219 7.3.5. Solid Decomposition Reactions ............................................................................................ 219 7.3.6. Solid Deposition Reactions ................................................................................................... 220 7.3.7. Gaseous Solid Catalyzed Reactions on the Particle Surface .................................................... 220 7.4. Reacting Channel Model ............................................................................................................... 220 7.4.1. Overview and Limitations ..................................................................................................... 220 7.4.2. Reacting Channel Model Theory ........................................................................................... 221 7.4.2.1. Flow Inside the Reacting Channel ................................................................................. 221 7.4.2.2. Outer Flow in the Shell ................................................................................................. 223 8. Non-Premixed Combustion ................................................................................................................. 225 8.1. Introduction ................................................................................................................................. 225 8.2. Non-Premixed Combustion and Mixture Fraction Theory ............................................................... 225 8.2.1. Mixture Fraction Theory ....................................................................................................... 226 8.2.1.1. Definition of the Mixture Fraction ................................................................................ 226 8.2.1.2. Transport Equations for the Mixture Fraction ................................................................ 228 8.2.1.3. The Non-Premixed Model for LES ................................................................................. 229 8.2.1.4. Mixture Fraction vs. Equivalence Ratio .......................................................................... 229 8.2.1.5. Relationship of Mixture Fraction to Species Mass Fraction, Density, and Temperature ..... 230 8.2.2. Modeling of Turbulence-Chemistry Interaction ..................................................................... 231 8.2.2.1. Description of the Probability Density Function ............................................................ 231 8.2.2.2. Derivation of Mean Scalar Values from the Instantaneous Mixture Fraction ................... 232 8.2.2.3. The Assumed-Shape PDF ............................................................................................. 233 8.2.2.3.1. The Double Delta Function PDF ........................................................................... 233 8.2.2.3.2. The -Function PDF ............................................................................................. 234 8.2.3. Non-Adiabatic Extensions of the Non-Premixed Model .......................................................... 235 8.2.4. Chemistry Tabulation ........................................................................................................... 237 8.2.4.1. Look-Up Tables for Adiabatic Systems ........................................................................... 237 8.2.4.2. 3D Look-Up Tables for Non-Adiabatic Systems .............................................................. 239 8.3. Restrictions and Special Cases for Using the Non-Premixed Model ................................................. 241 8.3.1. Restrictions on the Mixture Fraction Approach ...................................................................... 242 8.3.2. Using the Non-Premixed Model for Liquid Fuel or Coal Combustion ...................................... 244 8.3.3. Using the Non-Premixed Model with Flue Gas Recycle .......................................................... 245 8.3.4. Using the Non-Premixed Model with the Inert Model ............................................................ 246 8.3.4.1. Mixture Composition ................................................................................................... 246Release 14.0 - SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.

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ANSYS FLUENT Theory Guide 8.3.4.1.1. Property Evaluation ............................................................................................. 247 8.4.The Laminar Flamelet Models Theory ............................................................................................. 248 8.4.1. Restrictions and Assumptions ............................................................................................... 248 8.4.2. The Flamelet Concept ........................................................................................................... 248 8.4.2.1. Overview ..................................................................................................................... 248 8.4.2.2. Strain Rate and Scalar Dissipation ................................................................................. 250 8.4.2.3. Embedding Laminar Flamelets in Turbulent Flames ...................................................... 251 8.4.3. Flamelet Generation ............................................................................................................. 252 8.4.4. Flamelet Import ................................................................................................................... 252 8.5. The Steady Laminar Flamelet Model Theory ................................................................................... 254 8.5.1. Overview ............................................................................................................................. 254 8.5.2. Multiple Steady Flamelet Libraries ........................................................................................ 255 8.5.3. Steady Laminar Flamelet Automated Grid Refinement .......................................................... 255 8.5.4. Non-Adiabatic Steady Laminar Flamelets .............................................................................. 256 8.6. The Unsteady Laminar Flamelet Model Theory ............................................................................... 257 8.6.1. The Eulerian Unsteady Laminar Flamelet Model .................................................................... 257 8.6.1.1. Liquid Reactions .......................................................................................................... 259 8.6.2. The Diesel Unsteady Laminar Flamelet Model ....................................................................... 260 9. Premixed Combustion ......................................................................................................................... 263 9.1. Overview and Limitations ............................................................................................................. 263 9.1.1. Overview ............................................................................................................................. 263 9.1.2. Limitations ........................................................................................................................... 264 9.2. C-Equation Model Theory .............................................................................................................. 264 9.2.1. Propagation of the Flame Front ............................................................................................ 264 9.3. G-Equation Model Theory ............................................................................................................. 266 9.3.1. Numerical Solution of the G-equation ................................................................................... 267 9.4. Turbulent Flame Speed Models ..................................................................................................... 267 9.4.1. Zimont Turbulent Flame Speed Closure Model ...................................................................... 268 9.4.1.1. Zimont Turbulent Flame Speed Closure for LES ............................................................. 269 9.4.1.2. Flame Stretch Effect ..................................................................................................... 269 9.4.1.3. Gradient Diffusion ....................................................................................................... 270 9.4.1.4. Wall Damping .............................................................................................................. 271 9.4.2. Peters Flame Speed Model .................................................................................................... 271 9.4.2.1. Peters Flame Speed Model for LES ................................................................................ 272 9.5. Extended Coherent Flamelet Model Theory ................................................................................... 273 9.5.1. Closure for ECFM Source Terms ............................................................................................. 275 9.5.2.Turbulent Flame Speed in ECFM ............................................................................................ 278 9.5.3. LES and ECFM ...................................................................................................................... 278 9.6. Calculation of Properties ............................................................................................................... 281 9.6.1. Calculation of Temperature ................................................................................................... 281 9.6.1.1. Adiabatic Temperature Calculation ............................................................................... 282 9.6.1.2. Non-Adiabatic Temperature Calculation ....................................................................... 282 9.6.2. Calculation of Density .......................................................................................................... 282 9.6.3. Laminar Flame Speed ........................................................................................................... 283 9.6.4. Unburnt Density and Thermal Diffusivity ............................................................................... 283 10. Partially Premixed Combustion ........................................................................................................ 285 10.1. Overview .................................................................................................................................... 285 10.2. Limitations .................................................................................................................................. 285 10.3. Partially Premixed Combustion Theory ........................................................................................ 285 10.3.1. Calculation of Scalar Quantities ........................................................................................... 286 10.3.2. Laminar Flame Speed ......................................................................................................... 287 11. Composition PDF Transport .............................................................................................................. 289Release 14.0 - SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.

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ANSYS FLUENT Theory Guide 11.1. Overview and Limitations ............................................................................................................ 289 11.2. Composition PDF Transport Theory ............................................................................................. 289 11.3. The Lagrangian Solution Method ................................................................................................. 291 11.3.1. Particle Convection ............................................................................................................ 291 11.3.2. Particle Mixing ................................................................................................................... 292 11.3.2.1. The Modified Curl Model ............................................................................................ 292 11.3.2.2. The IEM Model ........................................................................................................... 293 11.3.2.3. The EMST Model ........................................................................................................ 293 11.3.2.4. Liquid Reactions ........................................................................................................ 293 11.3.3. Particle Reaction ................................................................................................................. 293 11.4. The Eulerian Solution Method ..................................................................................................... 295 11.4.1. Reaction ............................................................................................................................. 296 11.4.2. Mixing ................................................................................................................................ 296 11.4.3. Correction .......................................................................................................................... 296 11.4.4. Calculation of Composition Mean and Variance ................................................................... 297 12. Chemistry Acceleration ..................................................................................................................... 299 12.1. Overview and Limitations ............................................................................................................ 299 12.2. In-Situ Adaptive Tabulation (ISAT) ................................................................................................ 299 12.3. Chemistry Agglomeration ........................................................................................................... 301 12.3.1. Binning Algorithm .............................................................................................................. 303 12.4. Chemical Mechanism Dimension Reduction ................................................................................ 304 12.4.1. Selecting the Represented Species ...................................................................................... 305 13. Engine Ignition .................................................................................................................................. 307 13.1. Spark Model ................................................................................................................................ 307 13.1.1. Overview and Limitations ................................................................................................... 307 13.1.2. Spark Model Theory ............................................................................................................ 308 13.1.2.1. C-Equation Premixed Flame Model ............................................................................. 308 13.1.2.2. Other Combustion Models ......................................................................................... 308 13.1.3. ECFM Spark Model Variants ................................................................................................. 309 13.1.3.1. Boudier Model ........................................................................................................... 309 13.1.3.2. Teraji Model ............................................................................................................... 310 13.1.3.3. Zimont Model ............................................................................................................ 311 13.1.3.4. Constant Value Model ................................................................................................ 311 13.2. Autoignition Models ................................................................................................................... 311 13.2.1. Model Overview ................................................................................................................ 311 13.2.2. Model Limitations .............................................................................................................. 312 13.2.3. Ignition Model Theory ........................................................................................................ 312 13.2.3.1. Transport of Ignition Species ...................................................................................... 313 13.2.3.2. Knock Modeling ........................................................................................................ 313 13.2.3.2.1. Modeling of the Source Term ............................................................................. 313 13.2.3.2.2. Correlations ...................................................................................................... 314 13.2.3.2.3. Energy Release .................................................................................................. 314 13.2.3.3. Ignition Delay Modeling ............................................................................................. 315 13.2.3.3.1. Modeling of the Source Term ............................................................................. 315 13.2.3.3.2. Correlations ...................................................................................................... 316 13.2.3.3.3. Energy Release .................................................................................................. 316 13.3. Crevice Model ............................................................................................................................. 316 13.3.1. Overview ........................................................................................................................... 316 13.3.1.1. Model Parameters ...................................................................................................... 317 13.3.2. Limitations ......................................................................................................................... 318 13.3.3. Crevice Model Theory ......................................................................................................... 319 14. Pollutant Formation .......................................................................................................................... 321Release 14.0 - SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.

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ANSYS FLUENT Theory Guide 14.1. NOx Formation ........................................................................................................................... 321 14.1.1. Overview ........................................................................................................................... 321 14.1.1.1. NOx Modeling in ANSYS FLUENT ................................................................................ 321 14.1.1.2. NOx Formation and Reduction in Flames .................................................................... 322 14.1.2. Governing Equations for NOx Transport .............................................................................. 322 14.1.3. Thermal NOx Formation ...................................................................................................... 323 14.1.3.1. Thermal NOx Reaction Rates ...................................................................................... 323 14.1.3.2. The Quasi-Steady Assumption for [N] ......................................................................... 324 14.1.3.3. Thermal NOx Temperature Sensitivity ......................................................................... 324 14.1.3.4. Decoupled Thermal NOx Calculations ......................................................................... 325 14.1.3.5. Approaches for Determining O Radical Concentration ................................................ 325 14.1.3.5.1. Method 1: Equilibrium Approach ....................................................................... 325 14.1.3.5.2. Method 2: Partial Equilibrium Approach ............................................................. 326 14.1.3.5.3. Method 3: Predicted O Approach ....................................................................... 326 14.1.3.6. Approaches for Determining OH Radical Concentration .............................................. 326 14.1.3.6.1. Method 1: Exclusion of OH Approach ................................................................. 326 14.1.3.6.2. Method 2: Partial Equilibrium Approach ............................................................. 326 14.1.3.6.3. Method 3: Predicted OH Approach ..................................................................... 327 14.1.3.7. Summary ................................................................................................................... 327 14.1.4. Prompt NOx Formation ....................................................................................................... 327 14.1.4.1. Prompt NOx Combustion Environments ..................................................................... 327 14.1.4.2. Prompt NOx Mechanism ............................................................................................ 327 14.1.4.3. Prompt NOx Formation Factors .................................................................................. 328 14.1.4.4. Primary Reaction ....................................................................................................... 328 14.1.4.5. Modeling Strategy ..................................................................................................... 329 14.1.4.6. Rate for Most Hydrocarbon Fuels ................................................................................ 329 14.1.4.7. Oxygen Reaction Order .............................................................................................. 330 14.1.5. Fuel NOx Formation ............................................................................................................ 330 14.1.5.1. Fuel-Bound Nitrogen ................................................................................................. 330 14.1.5.2. Reaction Pathways ..................................................................................................... 331 14.1.5.3. Fuel NOx from Gaseous and Liquid Fuels .................................................................... 331 14.1.5.3.1. Fuel NOx from Intermediate Hydrogen Cyanide (HCN) ....................................... 331 14.1.5.3.1.1. HCN Production in a Gaseous Fuel ............................................................ 332 14.1.5.3.1.2. HCN Production in a Liquid Fuel ................................................................ 332 14.1.5.3.1.3. HCN Consumption .................................................................................... 333 14.1.5.3.1.4. HCN Sources in the Transport Equation ..................................................... 333 14.1.5.3.1.5. NOx Sources in the Transport Equation ..................................................... 334 14.1.5.3.2. Fuel NOx from Intermediate Ammonia (NH3) ..................................................... 334 14.1.5.3.2.1. NH3 Production in a Gaseous Fuel ............................................................. 335 14.1.5.3.2.2. NH3 Production in a Liquid Fuel ................................................................ 335 14.1.5.3.2.3. NH3 Consumption .................................................................................... 335 14.1.5.3.2.4. NH3 Sources in the Transport Equation ..................................................... 336 14.1.5.3.2.5. NOx Sources in the Transport Equation ..................................................... 336 14.1.5.3.3. Fuel NOx from Coal ........................................................................................... 337 14.1.5.3.3.1. Nitrogen in Char and in Volatiles ............................................................... 337 14.1.5.3.3.2. Coal Fuel NOx Scheme A ........................................................................... 337 14.1.5.3.3.3. Coal Fuel NOx Scheme B ........................................................................... 338 14.1.5.3.3.4. HCN Scheme Selection ............................................................................. 338 14.1.5.3.3.5. NOx Reduction on Char Surface ................................................................ 339 14.1.5.3.3.5.1. BET Surface Area .............................................................................. 340 14.1.5.3.3.5.2. HCN from Volatiles ........................................................................... 340 14.1.5.3.3.6. Coal Fuel NOx Scheme C ........................................................................... 340Release 14.0 - SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.

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ANSYS FLUENT Theory Guide 14.1.5.3.3.7. Coal Fuel NOx Scheme D ........................................................................... 341 14.1.5.3.3.8. NH3 Scheme Selection ............................................................................. 341 14.1.5.3.3.8.1. NH3 from Volatiles ........................................................................... 342 14.1.5.3.4. Fuel Nitrogen Partitioning for HCN and NH3 Intermediates ................................ 342 14.1.6. NOx Formation from Intermediate N2O ............................................................................... 343 14.1.6.1. N2O - Intermediate NOx Mechanism .......................................................................... 343 14.1.7. NOx Reduction by Reburning ............................................................................................. 344 14.1.7.1. Instantaneous Approach ............................................................................................ 344 14.1.7.2. Partial Equilibrium Approach ..................................................................................... 345 14.1.7.2.1. NOx Reduction Mechanism ............................................................................... 346 14.1.8. NOx Reduction by SNCR ..................................................................................................... 348 14.1.8.1. Ammonia Injection .................................................................................................... 349 14.1.8.2. Urea Injection ............................................................................................................ 350 14.1.8.3. Transport Equations for Urea, HNCO, and NCO ............................................................ 351 14.1.8.4. Urea Production due to Reagent Injection .................................................................. 352 14.1.8.5. NH3 Production due to Reagent Injection ................................................................... 352 14.1.8.6. HNCO Production due to Reagent Injection ................................................................ 353 14.1.9. NOx Formation in Turbulent Flows ...................................................................................... 353 14.1.9.1. The Turbulence-Chemistry Interaction Model ............................................................. 354 14.1.9.2. The PDF Approach ..................................................................................................... 354 14.1.9.3. The General Expression for the Mean Reaction Rate .................................................... 354 14.1.9.4. The Mean Reaction Rate Used in ANSYS FLUENT ......................................................... 354 14.1.9.5. Statistical Independence ............................................................................................ 355 14.1.9.6. The Beta PDF Option .................................................................................................. 355 14.1.9.7. The Gaussian PDF Option ........................................................................................... 356 14.1.9.8. The Calculation Method for the Variance .................................................................... 356 14.2. SOx Formation ............................................................................................................................ 357 14.2.1. Overview ........................................................................................................................... 357 14.2.1.1. The Formation of SOx ................................................................................................. 357 14.2.2. Governing Equations for SOx Transport ............................................................................... 358 14.2.3. Reaction Mechanisms for Sulfur Oxidation .......................................................................... 359 14.2.4. SO2 and H2S Production in a Gaseous Fuel ......................................................................... 361 14.2.5. SO2 and H2S Production in a Liquid Fuel ............................................................................. 361 14.2.6. SO2 and H2S Production from Coal ..................................................................................... 362 14.2.6.1. SO2 and H2S from Char .............................................................................................. 362 14.2.6.2. SO2 and H2S from Volatiles ........................................................................................ 362 14.2.7. SOx Formation in Turbulent Flows ....................................................................................... 362 14.2.7.1. The Turbulence-Chemistry Interaction Model ............................................................. 363 14.2.7.2. The PDF Approach ..................................................................................................... 363 14.2.7.3. The Mean Reaction Rate ............................................................................................. 363 14.2.7.4. The PDF Options ........................................................................................................ 363 14.3. Soot Formation ........................................................................................................................... 363 14.3.1. Overview and Limitations ................................................................................................... 363 14.3.1.1. Predicting Soot Formation ......................................................................................... 364 14.3.1.2. Restrictions on Soot Modeling ................................................................................... 364 14.3.2. Soot Model Theory ............................................................................................................. 364 14.3.2.1. The One-Step Soot Formation Model .......................................................................... 364 14.3.2.2. The Two-Step Soot Formation Model .......................................................................... 366 14.3.2.2.1. Soot Generation Rate ........................................................................................ 366 14.3.2.2.2. Nuclei Generation Rate ...................................................................................... 367 14.3.2.3. The Moss-Brookes Model ........................................................................................... 368 14.3.2.3.1. The Moss-Brookes-Hall Model ............................................................................ 370Release 14.0 - SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.

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ANSYS FLUENT Theory Guide 14.3.2.3.2. Soot Formation in Turbulent Flows .................................................................... 372 14.3.2.3.2.1. The Turbulence-Chemistry Interaction Model ............................................ 372 14.3.2.3.2.2. The PDF Approach .................................................................................... 372 14.3.2.3.2.3. The Mean Reaction Rate ........................................................................... 372 14.3.2.3.2.4. The PDF Options ....................................................................................... 372 14.3.2.3.3. The Effect of Soot on the Radiation Absorption Coefficient ................................. 372 14.4. Decoupled Detailed Chemistry Model ......................................................................................... 373 14.4.1. Overview ........................................................................................................................... 373 14.4.1.1. Limitations ................................................................................................................ 373 14.4.2. Decoupled Detailed Chemistry Model Theory ..................................................................... 373 15. Aerodynamically Generated Noise ................................................................................................... 375 15.1. Overview .................................................................................................................................... 375 15.1.1. Direct Method .................................................................................................................... 375 15.1.2. Integral Method Based on Acoustic Analogy ....................................................................... 376 15.1.3. Broadband Noise Source Models ........................................................................................ 377 15.2. Acoustics Model Theory .............................................................................................................. 377 15.2.1. The Ffowcs-Williams and Hawkings Model .......................................................................... 377 15.2.2. Broadband Noise Source Models ........................................................................................ 381 15.2.2.1. Proudmans Formula .................................................................................................. 381 15.2.2.2.The Jet Noise Source Model ........................................................................................ 381 15.2.2.3. The Boundary Layer Noise Source Model .................................................................... 384 15.2.2.4. Source Terms in the Linearized Euler Equations ........................................................... 385 15.2.2.5. Source Terms in Lilleys Equation ................................................................................ 386 16. Discrete Phase ................................................................................................................................... 387 16.1. Introduction ............................................................................................................................... 387 16.1.1. The Euler-Lagrange Approach ............................................................................................. 387 16.2. Particle Motion Theory ................................................................................................................ 388 16.2.1. Equations of Motion for Particles ........................................................................................ 388 16.2.1.1. Particle Force Balance ................................................................................................ 388 16.2.1.2. Inclusion of the Gravity Term ...................................................................................... 388 16.2.1.3. Other Forces .............................................................................................................. 389 16.2.1.4. Forces in Moving Reference Frames ............................................................................ 389 16.2.1.5. Thermophoretic Force ................................................................................................ 389 16.2.1.6. Brownian Force .......................................................................................................... 390 16.2.1.7. Saffmans Lift Force .................................................................................................... 391 16.2.2. Turbulent Dispersion of Particles ......................................................................................... 391 16.2.2.1. Stochastic Tracking .................................................................................................... 392 16.2.2.1.1. The Integral Time .............................................................................................. 392 16.2.2.1.2. The Discrete Random Walk Model ...................................................................... 393 16.2.2.1.3. Using the DRW Model ....................................................................................... 394 16.2.2.2. Particle Cloud Tracking ............................................................................................... 395 16.2.2.2.1. Using the Cloud Model ...................................................................................... 399 16.2.3. Integration of Particle Equation of Motion ........................................................................... 399 16.3. Laws for Drag Coefficients ........................................................................................................... 402 16.3.1. Spherical Drag Law ............................................................................................................. 402 16.3.2. Non-spherical Drag Law ..................................................................................................... 402 16.3.3. Stokes-Cunningham Drag Law ............................................................................................ 403 16.3.4. High-Mach-Number Drag Law ............................................................................................ 403 16.3.5. Dynamic Drag Model Theory .............................................................................................. 404 16.3.6. Dense Discrete Phase Model Drag Laws .............................................................................. 404 16.4. Laws for Heat and Mass Exchange ............................................................................................... 405 16.4.1. Inert Heating or Cooling (Law 1/Law 6) ............................................................................... 405Release 14.0 - SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.

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ANSYS FLUENT Theory Guide 16.4.2. Droplet Vaporization (Law 2) ............................................................................................... 408 16.4.2.1. Mass Transfer During Law 2 Diffusion Controlled Model ......................................... 408 16.4.2.2. Mass Transfer During Law 2 Convection/Diffusion Controlled Model ....................... 410 16.4.2.3. Defining the Vapor Pressure and Diffusion Coefficient ................................................. 410 16.4.2.4. Defining the Boiling Point and Latent Heat ................................................................. 411 16.4.2.5. Heat Transfer to the Droplet ....................................................................................... 412 16.4.3. Droplet Boiling (Law 3) ....................................................................................................... 413 16.4.4. Devolatilization (Law 4) ...................................................................................................... 414 16.4.4.1. Choosing the Devolatilization Model .......................................................................... 415 16.4.4.2.The Constant Rate Devolatilization Model ................................................................... 415 16.4.4.3. The Single Kinetic Rate Model .................................................................................... 415 16.4.4.4. The Two Competing Rates (Kobayashi) Model ............................................................. 416 16.4.4.5. The CPD Model .......................................................................................................... 417 16.4.4.5.1. General Description .......................................................................................... 417 16.4.4.5.2. Reaction Rates .................................................................................................. 418 16.4.4.5.3. Mass Conservation ............................................................................................ 419 16.4.4.5.4. Fractional Change in the Coal Mass .................................................................... 419 16.4.4.5.5. CPD Inputs ........................................................................................................ 421 16.4.4.5.6. Particle Swelling During Devolatilization ............................................................ 422 16.4.4.5.7. Heat Transfer to the Particle During Devolatilization ........................................... 422 16.4.5. Surface Combustion (Law 5) ............................................................................................... 423 16.4.5.1. The Diffusion-Limited Surface Reaction Rate Model .................................................... 424 16.4.5.2. The Kinetic/Diffusion Surface Reaction Rate Model ..................................................... 424 16.4.5.3. The Intrinsic Model .................................................................................................... 425 16.4.5.4. The Multiple Surface Reactions Model ........................................................................ 427 16.4.5.4.1. Limitations ........................................................................................................ 427 16.4.5.5. Heat and Mass Transfer During Char Combustion ....................................................... 428 16.4.6. Multicomponent Particle Definition (Law 7) ........................................................................ 428 16.4.6.1. Raoults Law .............................................................................................................. 430 16.4.6.2. Peng-Robinson Real Gas Model ....